18.S66 PROBLEMS #5
Spring 2003
Let us define the nth Catalan number Cn by
1
2n
, n ≥ 0.
Cn =
n+1 n
(5)
Thus (C0 , C1 , . . .) = (1, 1, 2, 5, 14, 42, 132, 429, . . .). There are a huge number
of combinatorial interpretations of these numbers; 66 appear in Exercise 6.19
of R. Stanley, Enumerative Combinatorics, vol. 2. This exercise (as well as
some related ones) is available at www-math.mit.edu/∼rstan/ec, and an addendum with many more interpretations may be found at the same website.
We give here a subset of these interpretations that are the most fundamental
or most interesting. Problem 143 is perhaps the easiest one to show bijectively is counted by (5). All your other proofs should be bijections with
previously shown “Catalan sets.” Each interpretation is illustrated by the
case n = 3, which hopefully will make any undefined terms clear. Needless
to say, you should not hand in a problem whose solution you have obtained
from an outside source (except reasonable collaboration with other students
in the course).
132. [1.5] triangulations of a convex (n + 2)-gon into n triangles by n − 1
diagonals that do not intersect in their interiors
133. [1.5] binary parenthesizations of a string of n + 1 letters
(xx · x)x
x(xx · x)
(x · xx)x
x(x · xx)
r
@
@r
r
xx · xx
134. binary trees with n vertices
r
@ r
@
@r
r
r
@
@r
r
31
r
r
r
r
@
@r
135. [1.5] plane binary trees with 2n + 1 vertices (or n + 1 endpoints) (A
plane binary tree is a binary tree for which every vertex is either an
endpoint or has two children.)
136. [2] plane trees with n+1 vertices (A plane tree is a rooted tree for which
the subtrees of every vertex are linearly ordered from left to right.)
r
r
r
r
r
r
r
@
@r
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r
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@
@r
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@
@r
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r
r
@r
r@
137. [1.5] lattice paths from (0, 0) to (n, n) with steps (0, 1) or (1, 0), never
rising above the line y = x
r
r
r r
r r r
r
r
r
r r r r
r
r
r r r
r r
r
r r
r
r r r
r
r r
r r
r r
138. [1] Dyck paths from (0, 0) to (2n, 0), i.e., lattice paths with steps (1, 1)
and (1, −1), never falling below the x-axis
139. [2.5] (unordered) pairs of lattice paths with n + 1 steps each, starting
at (0, 0), using steps (1, 0) or (0, 1), ending at the same point, and only
intersecting at the beginning and end
r
r
r
r
r
r
r
r
r
r
r
r
r
r
r
r
r
r
r
32
r
r
r
r
r
r
r
r
r
r
r
r
r
r
r
r
r
r
r
r
r
140. [1.5] n nonintersecting chords joining 2n points on the circumference
of a circle
r
r
r
r
r
r
r
r
r
@
@r
r
r r
r AA @
@r
@
@r Ar
r
r
r
r
r
@
@r
r
r
r
r
r r
r
141. [2] ways of drawing in the plane n + 1 points lying on a horizontal line
L and n arcs connecting them such that (α) the arcs do not pass below
L, (β) the graph thus formed is a tree, (γ) no two arcs intersect in their
interiors (i.e., the arcs are noncrossing), and (δ) at every vertex, all the
arcs exit in the same direction (left or right)
142. [2.5] ways of drawing in the plane n + 1 points lying on a horizontal
line L and n arcs connecting them such that (α) the arcs do not pass
below L, (β) the graph thus formed is a tree, (γ) no arc (including its
endpoints) lies strictly below another arc, and (δ) at every vertex, all
the arcs exit in the same direction (left or right)
143. [1] sequences of n 1’s and n −1’s such that every partial sum is nonnegative (with −1 denoted simply as − below)
111−−−
11−1−−
11−−1−
1−11−−
1−1−1−
144. [1] sequences 1 ≤ a1 ≤ · · · ≤ an of integers with ai ≤ i
111
112
113
122
123
145. [2] sequences a1 , a2 , . . . , an of integers such that a1 = 0 and 0 ≤ ai+1 ≤
ai + 1
000 001 010 011 012
146. [1.5] sequences a1 , a2 , . . . , an−1 of integers such that ai ≤ 1 and all
partial sums are nonnegative
0, 0
0, 1
1, −1
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1, 0
1, 1
147. [1.5] sequences a1 , a2 , . . . , an of integers such that ai ≥ −1, all partial
sums are nonnegative, and a1 + a2 + · · · + an = 0
0, 1, −1
0, 0, 0
1, 0, −1
1, −1, 0
2, −1, −1
148. [1.5] Sequences of n − 1 1’s and any number of −1’s such that every
partial sum is nonnegative
1, 1
1, 1, −1
1, −1, 1
1, 1, −1, −1
1, −1, 1, −1
149. [2.5] Sequences a1 a2 · · · an of nonnegative integers such that aj = #{i :
i < j, ai < aj } for 1 ≤ j ≤ n
000
002
010
011
012
150. [2.5] Pairs (α, β) of compositions of n with the same number of parts,
such that α ≥ β (dominance order, i.e., α1 + · · · + αi ≥ β1 + · · · + βi
for all i)
(111, 111) (12, 12) (21, 21) (21, 12) (3, 3)
151. [2] permutations a1 a2 · · · a2n of the multiset {12 , 22 , . . . , n2 } such that:
(i) the first occurrences of 1, 2, . . . , n appear in increasing order, and
(ii) there is no subsequence of the form αβαβ
112233
112332
122331
123321
122133
152. [2.5] permutations a1 a2 · · · an of [n] with longest decreasing subsequence
of length at most two (i.e., there does not exist i < j < k, ai > aj > ak ),
called 321-avoiding permutations
123
213
132
312
231
153. [2] permutations a1 a2 · · · an of [n] for which there does not exist i <
j < k and aj < ak < ai (called 312-avoiding permutations)
123
132
213
34
231
321
154. [2] permutations w of [2n] with n cycles of length two, such that the
product (1, 2, . . . , 2n) · w has n + 1 cycles
(1, 2, 3, 4, 5, 6)(1, 2)(3, 4)(5, 6) = (1)(2, 4, 6)(3)(5)
(1, 2, 3, 4, 5, 6)(1, 2)(3, 6)(4, 5) = (1)(2, 6)(3, 5)(4)
(1, 2, 3, 4, 5, 6)(1, 4)(2, 3)(5, 6) = (1, 3)(2)(4, 6)(5)
(1, 2, 3, 4, 5, 6)(1, 6)(2, 3)(4, 5) = (1, 3, 5)(2)(4)(6)
(1, 2, 3, 4, 5, 6)(1, 6)(2, 5)(3, 4) = (1, 5)(2, 4)(3)(6)
155. [2.5] pairs (u, v) of permutations of [n] such that u and v have a total
of n + 1 cycles, and uv = (1, 2, . . . , n)
(1)(2)(3) · (1, 2, 3)
(1, 2, 3) · (1)(2)(3)
(1, 3)(2) · (1)(2, 3)
(1, 2)(3) · (1, 3)(2)
(1)(2, 3) · (1, 2)(3)
156. [2] noncrossing partitions of [n], i.e., partitions of [n] such that if a, c
appear in a block B and b, d appear in a block B 0 , where a < b < c < d,
then B = B 0
123
12−3
13−2
23−1
1−2−3
(The unique partition of [4] that isn’t noncrossing is 13−24.)
157. [2.5] noncrossing partitions of [2n + 1] into n + 1 blocks, such that no
block contains two consecutive integers
137−46−2−5 1357−2−4−6 157−24−3−6 17−246−3−5 17−26−35−4
158. [2.5] nonnesting partitions of [n], i.e., partitions of [n] such that if a, e
appear in a block B and b, d appear in a different block B 0 where
a < b < d < e, then there is a c ∈ B satisfying b < c < d
123
12−3
13−2
23−1
1−2−3
(The unique partition of [4] that isn’t nonnesting is 14−23.)
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159. [2.5] nonisomorphic n-element posets (i.e., partially ordered sets) with
no induced subposet isomorphic to 2 + 2 or 3 + 1, where a + b denotes
the disjoint union of an a-element chain and a b-element chain
r
r
r
r
r
r
r
@
@r
r
r
r
@
@r
r
r
r
160. [2] relations R on [n] that are reflexive (iRi), symmetric (iRj ⇒ jRi),
and such that if 1 ≤ i < j < k ≤ n and iRk, then iRj and jRk (in the
example below we write ij for the pair (i, j), and we omit the pairs ii)
∅ {12, 21} {23, 32} {12, 21, 23, 32} {12, 21, 13, 31, 23, 32}
161. [1.5] ways to stack coins in the plane, the bottom row consisting of n
consecutive coins
mmm
m
mmm
m
mmm
mm
mmm
m
mm
mmm
162. [2.5] n-tuples (a1 , a2 , . . . , an ) of integers ai ≥ 2 such that in the sequence
1a1 a2 · · · an 1, each ai divides the sum of its two neighbors
14321
13521
13231
12531
12341
163. [3] n-element subsets S of N × N such that if (i, j) ∈ S then i ≥ j and
there is a lattice path from (0, 0) to (i, j) with steps (0, 1), (1, 0), and
(1, 1) that lies entirely inside S
{(0, 0), (1, 0), (2, 0)} {(0, 0), (1, 0), (1, 1)} {(0, 0), (1, 0), (2, 1)}
{(0, 0), (1, 1), (2, 1)} {(0, 0), (1, 1), (2, 2)}
164. [3] positive integer sequences a1 , a2 , . . . , an+2 for which there exists an
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1
1
1
1
1
3
2
2
5
1
1
3
2
1
1
1
4
2
3
5
3
2
1
2
1
1
1
3
2
5
5
2
1
1
5
4
3
5
1
1
1
1
1
3
4
1
1
3
2
4
1
1
1
1
3
5
1
1
1
3
2
2
3
1
2
1
3
2
1
1
4
3
5
1
5
2
1
Figure 1: The frieze pattern corresponding to the sequence (1, 3, 2, 1, 5, 1, 2, 3)
integer array (necessarily with n + 1 rows)
1
1
a1
b1
···
1
a2
b2
1
1
···
a3
an+2
···
b3
···
1
a1
bn+2
···
b1
1
···
a2
bn−2
·
·
r1
·
r2
1
···
r3
1
rn+2
···
1
r1
1
(6)
r
such that any four neighboring entries in the configuration s t satu
isfy st = ru + 1 (an example of such an array for (a1 , . . . , a8 ) =
(1, 3, 2, 1, 5, 1, 2, 3) (necessarily unique) is given by Figure 1):
12213
22131
165. [3] n-tuples (a1 , . . . an ) of
matrix

a1 1
 1 a2

 0 1







 0 0
0 0
21312
13122
31221
positive integers such that the tridiagonal

0 0 · · ·
0
0
1 0 · · ·
0
0 

a3 1 · · ·
0
0 


·


·


·

0 0 · · · an−1 1 
0 0 · · ·
1
an
37
1
an−1
is positive definite with determinant one
Note. A real matrix A is positive definite if it is symmetric and every
eigenvalue is positive; equivalently, A is symmetric and every leading
principal minor is positive. A leading principal minor is the determinant of a square submatrix that fits into the upper left-hand corner of
A.
131 122 221 213 312
166. [2] Vertices of height n−1 of the tree T defined by the property that the
root has degree 2, and if the vertex x has degree k, then the children
of x have degrees 2, 3, . . . , k + 1
167. [2.5] Subsets S of N such that 0 ∈ S and such that if i ∈ S then
i + n, i + n + 1 ∈ S
N,
N − {1},
N − {2},
N − {1, 2},
N − {1, 2, 5}
168. [2] Ways to write (1, 1, . . . , 1, −n) ∈ Zn+1 as a sum of vectors ei −
ei+1 and ej − en+1 , without regard to order, where ek is the kth unit
coordinate vector in Zn+1 :
(1, −1, 0, 0) + 2(0, 1, −1, 0) + 3(0, 0, 1, −1)
(1, 0, 0, −1) + (0, 1, −1, 0) + 2(0, 0, 1, −1)
(1, −1, 0, 0) + (0, 1, −1, 0) + (0, 1, 0, −1) + 2(0, 0, 1, −1)
(1, −1, 0, 0) + 2(0, 1, 0, −1) + (0, 0, 1, −1)
(1, 0, 0, −1) + (0, 1, 0, −1) + (0, 0, 1, −1)
169. [1.5] tilings of the staircase shape (n, n − 1, . . . , 1) with n rectangles
such that each rectangle contains a square at the end of some row
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170. [2] n × n N-matrices M = (mij ) where mij = 0 unless i = n or i = j or
i = j − 1, with row and column sum vector (1, 2, . . . , n)

 
 
 
 

1 0 0
0 1 0
1 0 0
1 0 0
0 1 0
 0 2 0   0 1 1   0 1 1   0 0 2   0 0 2 
0 0 3
1 0 2
0 1 2
0 2 1
1 1 1
This concludes the list of objects counted by Catalan numbers. A few
more problems related to Catalan numbers are the following.
171. (*) We have
n
X
C2k C2(n−k) = 4n Cn .
k=0
172. (*) An intriguing variation of Problem 170 above is the following. A
bijective proof would be of great interest. Let g(n) denote the number
of n × n N-matrices M = (mij ) where mij = 0 if i > j + 1, with row
and column sum vector 1, 3, 6, . . . , n+1
. For instance, when n = 2
2
there are the two matrices
1 0
0 1
.
0 3
1 2
Then g(n) = C1 C2 · · · Cn .
173. [2] (compare with Problem 168) Let f (n) be the number of ways to
write the vector
n+1
∈ Zn+1
1, 2, 3, . . . , n, −
2
as a sum of vectors ei − ej , 1 ≤ i < j ≤ n + 1, without regard to
order, where ek is the kth unit coordinate vector in Zn+1 . For instance, when n = 2 there are the two ways (1, 2, −3) = (1, 0, −1) +
2(0, 1, −1) = (1, −1, 0) + 3(0, 1, −1). Assuming Problem 172, show
that f (n) = C1 C2 · · · Cn .
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174. [2.5] The Narayana numbers N (n, k) are defined by
n
1 n
.
N (n, k) =
k−1
n k
Let Xnk be the set of all sequences w = w1 w2 · · · w2n of n 1’s and n
−1’s with all partial sums nonnegative, such that
k = #{j : wj = 1, wj+1 = −1}.
Show that N (n, k) = #Xnk . Hence by Problem 143, there follows
n
X
N (n, k) = Cn .
k=1
One therefore says that the Narayana numbers are a refinement of
the Catalan numbers. There are many other interesting refinements of
Catalan numbers, but we won’t consider them here.
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